51
The Discontinuous Galerkin Method and Hyperbolic Problems Trial Lecture 2014-06-27 Martin Lilleeng Sætra
The Discontinuous Galerkin Method
and Hyperbolic Problems
Trial Lecture
2014-06-27
Martin Lilleeng Sætra
Outline
• Introduction
– Hyperbolic Conservation Laws
– Numerical Methods for PDEs
• The Discontinuous Galerkin (DG) Method
– Introduction in 1D
– Extension to higher order and higher dimensions
– Parallel Computation
• Summary and Further Reading
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Boris Galerkin
1871–1945
Introduction
1. Continuous problem
2. Impose spatial discretization
3. Approximate solution on each cell
4. Compute fluxes across cell interfaces
5. Evolve solution in time
6. (Do all this as efficiently as possible)
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The DG Method – Short History
• DG was first proposed by Reed and Hill in 1973
𝜎𝑢 + 𝛻 ∗ 𝑎𝑢 = 𝑓
• First analysis (LeSaint and Raviart,1974) showing order 𝑘accurate in general and often 𝑘 + 1 accurate for smooth
solutions (𝑘 polynomial degree of piecewise approximation)
• Sharp analysis (Johnson, 1986) showed 𝑘 +1
2-order accurate
• However, the schemes did not enjoy much use early on
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The DG Method – Short History
• Extension from scalar conservation laws in late 1980s to
systems in late 1990s (Cockburn and Shu)
• Development of limiters and Runge-Kutta DG methods for
nonlinear conservation laws
• New application areas, e.g., Maxwell’s equations, magneto-
hydrodynamics, water waves, and elasticity
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The DG Method – Short History
• The last decade has seen an explosion in activities
– Higher order problems
– Hamilton-Jacobi equations
– Non-coercive problems and spectral accuracy
– Adaptive solution techniques
– Improved solvers
– Advanced time-integration methods
– Large scale production codes
– Etc.
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Classes of PDEs
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Elliptic
• Infinite speed of
propagation
• All cells coupled
• Must assemble and solve
a linear system
• Prototypical example:
Laplace’s Equation
Parabolic
• Typically describe
diffusion phenomena
• Can use explicit time
stepping methods
• Prototypical example:
The Heat Equation
Hyperbolic Conservation Laws
• Fundamental in nature
• Conservation laws arise in a multitude of
different scientific areas
• For example:– Magneto-hydrodynamics
– Elastic waves
– Acoustic waves
– Gravitational waves
– Fluid dynamics (Euler eq’s, etc.)
• Discontinuities propagate without being
smoothed, e.g., shock waves
• Solution propagates with a finite speed
– Enables explicit time-stepping schemes
– Finite domains of influence and dependence
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Hyperbolic Conservation Laws
A conservation law states that the sum of a particular quantity within an isolated
system does not change as the system evolves in time, e.g., mass, momentum,
and energy
A general nonlinear hyperbolic conservation law in 1D can be written in differential
form as 𝑢𝑡 + 𝐹 𝑢 𝑥 = 𝑆(𝑢),
in which 𝑢(𝑥, 𝑡) is a vector of conserved variables, 𝐹 is the flux vector, and 𝑆 is a
source vector
Solutions generally defined in a weak sense only
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Hyperbolic Conservation Laws
• The Shallow Water Equations (written in 1D)
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Fresnel visualization (2D)
Hyperbolic Conservation Laws
• The Euler Equations of compressible gas dynamics (written in 1D)
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Schlieren visualization (2D)
Numerically Solving PDEs
In general, we cannot find an analytical solution
In the following we consider the 1D scalar conservation law
in which 𝑓 𝑢 is the flux function and s 𝑥, 𝑡 is a source function
We will now briefly investigate three classical methods for solving this
PDE numerically and list some benefits and problems associated with
each method
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𝜕𝑢
𝜕𝑡+
𝜕𝑓
𝜕𝑥= 𝑠, 𝑥 ∈ Ω,
Numerically Solving PDEs
• To construct a numerical method for solving PDEs we need to
consider
– How to represent the solution 𝑢(𝑥, 𝑡) by an approximate solution 𝑢ℎ(𝑥, 𝑡)?
– In which sense will the approximate solution 𝑢ℎ(𝑥, 𝑡) satisfy the PDE?
• These two choices separate and define the properties of different
numerical methods
• Bottom line is that we need ways to
– Generate a system of algebraic equations from the well-posed PDE
– Incorporate boundary conditions
– Solve the system of equations while minimizing unavoidable errors that are
introduced in the process
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Finite Difference Methods (FDM)
• Domain is represented by a set of points
• Solution approximated by function values at these points
• The equation is satisfied in a pointwise manner
𝑑𝑢ℎ(𝑥𝑗 , 𝑡)
𝑑𝑡+
𝑓ℎ 𝑥𝑗+1, 𝑡 − 𝑓ℎ 𝑥𝑗−1, 𝑡
ℎ𝑗 + ℎ𝑗−1= 𝑠(𝑥𝑗 , 𝑡)
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𝑥𝑗−1 𝑥𝑗 𝑥𝑗+1
ℎ𝑗
Finite Difference Methods (FDM)
• Main benefits
– Simple to understand
– Straightforward implementation on structured meshes
– Method is local and can be made explicit in time
– Simple techniques for local adaptivity (upwinding)
– Extensive body of theoretical and practical work on these methods since
the 1960s
• Main problems
– Implementation complexity increases for irregularly shaped geometry
– Less well-suited for problems with discontinuities
– Grid smoothness requirements
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Finite Volume Methods (FVM)
• Domain is represented by non-overlapping cells
• The local approximation is a cell average
𝑥
𝑗−12
𝑥𝑗+
12 𝑢ℎ 𝑥 𝑑𝑥 = ℎ𝑗 𝑢𝑗
• The equation is satisfied on conservation form
ℎ𝑗
𝑑 𝑢𝑗
𝑑𝑡+ 𝑓
𝑗+12
− 𝑓𝑗−
12
= ℎ𝑗 𝑠𝑗2014-06-27
𝑢𝑗
𝑢𝑗+1
ℎ𝑗𝑥𝑗−
1
2
𝑥𝑗+
12
Finite Volume Methods (FVM)
• Not straightforward to evaluate the fluxes on cell interfaces
• Numerical flux functions are generally used:
𝑓𝑗+
12
= 𝐹(𝑢𝑗+
12
− , 𝑢𝑗+
12
+ )
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𝑥𝑗+
1
2
𝑢𝑗
𝑢𝑗+1
Finite Volume Methods (FVM)
• Main benefits
– Robust and fast due to locality
– Support resolution of complex geometries
– Well-suited for hyperbolic conservation laws (local upwinding)
– Method is local and can be made explicit in time
– Method is locally conservative
– Extensive theoretical framework since the 1970s
• Main problems
– Difficult to achieve high-order accuracy in a straightforward way on general
grids due to requirement for extended stencils (reconstruction problem)
– Grid smoothness requirements
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Finite Element Methods (FEM)
• Domain is represented by non-overlapping elements
• The solution is defined in a non-local manner, using piecewise
continuous polynomials:
𝑢ℎ 𝑥 =
𝑗=1
𝐽
𝑢 𝑥𝑗 𝑁𝑗(𝑥)
• The equation is satisfied globally:
Ω
𝜕𝑢ℎ
𝜕𝑡+
𝜕𝑓ℎ
𝜕𝑥− 𝑠ℎ 𝑁𝑘 𝑥 𝑑𝑥 = 0
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Finite Element Methods (FEM)
• Main benefits
– Robust
– Systematic implementation on unstructured meshes
– High-order accuracy can be combined with complex geometries
– Well-suited for elliptic problems (global statement)
– Extensive theoretical framework since the 1970s
• Main problems
– Standard Galerkin not well-suited for problems with direction (global
statement), but there are techniques to handle such problems, like Petrov-
Galerkin, streamline diffusion, etc.
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Summary of Numerical Methods
• Want a scheme that combines:– The high-order element of FEM
– The geometric flexibility of FEM and FVM
– The local statement on the equation for FVM
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The Discontinuous Galerkin Finite Element Method
Complex
geometries
High-order
accuracy and
𝒉𝒑-adaptivity
Explicit semi-
discrete form
Conservation
laws
Elliptic
problems
FDM ✖ ✔ ✔ ✔ ✔
FVM ✔ ✖ ✔ ✔ (✔)
FEM ✔ ✔ ✖ (✔) ✔
DG ✔ ✔ ✔ ✔ (✔)
The Discontinuous Galerkin Method
We start with a 1D homogeneous scalar conservation law
𝑢𝑡 + 𝑓 𝑢 𝑥 = 0
Multiply with a test function 𝑣 and integrate over some cell 𝐼𝑗 = [𝑥𝑗−
1
2
, 𝑥𝑗+
1
2
]
𝐼𝑗
𝑢𝑡𝑣 + 𝑓 𝑢 𝑥𝑣𝑑𝑥 = 0
Then integrate by parts
𝐼𝑗
𝑢𝑡𝑣𝑑𝑥 − 𝐼𝑗
𝑓(𝑢) 𝑣𝑥𝑑𝑥 + 𝑓(𝑢𝑗+
12) 𝑣
𝑗+12
− 𝑓(𝑢𝑗−
12)𝑣
𝑗−12
= 0
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The Discontinuous Galerkin Method
Next, we assume that both the solution 𝑢 and the test function 𝑣 comes from a
finite dimensional approximation space 𝑉ℎ, usually the space of piecewise
polynomials of degree up to 𝑘:
𝑉ℎ = {𝑣: 𝑣 𝐼𝑗
∈ 𝑃𝑘 𝐼𝑗 , 𝑗 = 1, … , 𝑁}
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0 1
ℎ𝑗
For example 𝑃2 = {1, 𝑥, 𝑥2}
Assume unit cell to simplify
the further derivation𝐼𝑗
Let us start with the simple basis of 𝑃0 = {1}:
𝐼𝑗
𝑢𝑡𝑣𝑑𝑥 − 𝐼𝑗
𝑓(𝑢) 𝑣𝑥𝑑𝑥 + 𝑓(𝑢𝑗+
12) 𝑣
𝑗+12
− 𝑓(𝑢𝑗−
12)𝑣
𝑗−12
= 0
𝐼𝑗
𝑢𝑡1𝑑𝑥 − 𝐼𝑗
𝑓(𝑢) 0𝑑𝑥 + 𝑓(𝑢𝑗+
12) 𝑣
𝑗+12
− 𝑓(𝑢𝑗−
12)𝑣
𝑗−12
= 0
𝐼𝑗
𝑢𝑡𝑑𝑥 + 𝑓(𝑢𝑗+
12)𝑣
𝑗+12
− 𝑓(𝑢𝑗−
12)𝑣
𝑗−12
= 0
Example: DG(0)
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𝑢𝑗
𝑓𝑗+
12
= 𝐹(𝑢𝑗+
12
− , 𝑢𝑗+
12
+ )
This is nothing else than the finite volume method
Example: DG(1)
We try again, using a different basis, P1 = {1, x}:
𝐼𝑗
𝑢𝑡𝑣𝑑𝑥 − 𝐼𝑗
𝑓(𝑢) 𝑣𝑥𝑑𝑥 + 𝑓(𝑢𝑗+
12) 𝑣
𝑗+12
− 𝑓(𝑢𝑗−
12)𝑣
𝑗−12
= 0
𝑣 = 1
𝐼𝑗
𝑢𝑡𝑑𝑥 + 𝑓(𝑢𝑗+
12)𝑣
𝑗+12
− 𝑓(𝑢𝑗−
12)𝑣
𝑗−12
= 0
𝑣 = 𝑥
𝐼𝑗
𝑢𝑡𝑥𝑑𝑥 − 𝐼𝑗
𝑓(𝑢) 𝑑𝑥 + 𝑓(𝑢𝑗+
12) 𝑣
𝑗+12
− 𝑓(𝑢𝑗−
12)𝑣
𝑗−12
= 0
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New term
Example: DG(1)
By rearranging and inserting
𝑢 𝑥, 𝑡 = 𝑎0 𝑡 ∗ 1 + 𝑎1 𝑡 ∗ 𝑥
we get
𝑣 = 1𝑑𝑎0
𝑑𝑡 𝐼𝑗
1𝑑𝑥 +𝑑𝑎1
𝑑𝑡 𝐼𝑗
𝑥𝑑𝑥 = −𝑓(𝑢𝑗+
12)𝑣
𝑗+12
+𝑓(𝑢𝑗−
12)𝑣
𝑗−12
𝑣 = 𝑥
𝑑𝑎0
𝑑𝑡 𝐼𝑗
𝑥𝑑𝑥 +𝑑𝑎1
𝑑𝑡 𝐼𝑗
𝑥2𝑑𝑥 = 𝐼𝑗
𝑓(𝑎0 𝑡 + 𝑎1 (𝑡)𝑥) 𝑑𝑥 − 𝑓(𝑢𝑗+
12)𝑣
𝑗+12
+𝑓(𝑢𝑗−
12)𝑣
𝑗−12
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1
2
1
3
1
21
Assuming
unit cell
1 0
01
Example: DG(1)
𝑣 = 𝑥
1
2
𝑑𝑎0
𝑑𝑡+
1
3
𝑑𝑎1
𝑑𝑡=
𝐼𝑗
𝑓(𝑎0 + 𝑎1𝑥) 𝑑𝑥 − 𝑓(𝑢𝑗+
12)
For 1D we can simply use:
Approximate with an
internal quadrature
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Trapezoidal
≈1
2[𝑓(𝑢
𝑗−12) + 𝑓(𝑢
𝑗+12)]
=1
2[𝑓(𝑎0) + 𝑓(𝑎0 + 𝑎1)]
Simpsons
≈1
6[𝑓(𝑢
𝑗−12) + 4𝑓(
1
2[𝑢
𝑗−12
+ 𝑢𝑗+
12]) + 𝑓(𝑢
𝑗+12)]
=1
6[𝑓(𝑎0) + 4𝑓(
1
2[2𝑎0 + 𝑎1]) + 𝑓(𝑎0 + 𝑎1)]
𝑢𝑗−
12
𝑢𝑗+
12
• Now we have an explicit spatial discretization per cell
• Can view each cell as a separate entity that needs boundary
data from its neighbors
• Next, find the fluxes across cell interfaces
Example: DG(1) – Interfaces
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𝑥𝑗−1 𝑥𝑗 𝑥𝑗+1
Example: DG(1) – Numerical Flux
𝑣 = 𝑥
1
2
𝑑𝑎0
𝑑𝑡+
1
3
𝑑𝑎1
𝑑𝑡=
1
2[𝑓(𝑎0
𝑛 ) + 𝑓(𝑎0𝑛 + 𝑎1
𝑛 )] − 𝑓(𝑢𝑗+
12)
In general we
approximate the
flux with a
numerical flux
function
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𝑓(𝑢𝑗+
12) ≈ 𝐹
𝑗+12(𝑢
𝑗+12
− , 𝑢𝑗+
12
+ )
𝑢−
𝑢+
𝐼𝑗 𝐼𝑗+1
Choice of Numerical Flux Function
• Builds on extensive knowledge from development of numerical fluxes
for FVM schemes
• Problem dependent
• Some examples:
Lax-Friedrichs:
𝐹𝐿𝐹 𝑢−, 𝑢+ =1
2𝑓 𝑢− + 𝑓 𝑢+ − 𝛼 𝑢+ − 𝑢− , 𝛼 = max
𝑢|𝑓′(𝑢)|
Godunov:
𝐹𝐺 𝑢−, 𝑢+ = min
𝑢−≤𝑢≤𝑢+𝑓 𝑢 , 𝑖𝑓 𝑢− < 𝑢+
max𝑢+≤𝑢≤𝑢−
𝑓 𝑢 , 𝑖𝑓 𝑢− ≥ 𝑢+
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𝑢−
𝑢+
Semi-Discrete Formulation
After inserting quadrature and numerical flux function, we are now left with two
ODEs to solve in time:
𝑣 = 1𝑑𝑎0
𝑑𝑡+
1
2
𝑑𝑎1
𝑑𝑡= −𝐹
𝑗+12(𝑢
𝑗+12
− , 𝑢𝑗+
12
+ )
𝑣 = 𝑥
1
2
𝑑𝑎0
𝑑𝑡+
1
3
𝑑𝑎1
𝑑𝑡=
1
2[𝑓(𝑎0
𝑛 ) + 𝑓(𝑎0𝑛 + 𝑎1
𝑛 )] − 𝐹𝑗+
12(𝑢
𝑗+12
− , 𝑢𝑗+
12
+ )
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Explicit Discontinous Galerkin
• Hyperbolic PDEs
– Finite speed of propagation
– Solution will propagate a certain distance over a period of time, Δ𝑡
• Locality of DG Method
• Use explicit time stepping/integration
• In the following we will use simple Euler time integration to
evolve the solution in time
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Time Integration
We then use an explicit solver in time, simple forward Euler in this case:
𝑣 = 1𝑎0
𝑛+1 − 𝑎0𝑛
Δ𝑡+
1
2
𝑎1𝑛+1 − 𝑎1
𝑛
Δ𝑡= −𝐹
𝑗+12(𝑢
𝑗+12
− , 𝑢𝑗+
12
+ )
𝑣 = 𝑥
1
2
𝑎0𝑛+1 − 𝑎0
𝑛
Δ𝑡+
1
3
𝑎1𝑛+1 − 𝑎1
𝑛
Δ𝑡=
1
2[𝑓(𝑎0
𝑛 ) + 𝑓(𝑎0𝑛 + 𝑎1
𝑛 )] − 𝐹𝑗+
12(𝑢
𝑗+12
− , 𝑢𝑗+
12
+ )
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𝑅(𝑢)
Compute 𝑅 𝑢
and solve linear system for 𝑎0𝑛+1 and 𝑎1
𝑛+1
Higher Order in Time
• For higher order in time, we can use, e.g., Runge-Kutta (RK)
• For the semi-discrete scheme:𝑑𝑢
𝑑𝑡= 𝑅 𝑢 ,
in which 𝑅 𝑢 is the spatial DG discretization, the 3rd order RK is:
𝑢(1) = 𝑢𝑛 + Δ𝑡𝑅 𝑢𝑛
𝑢(2) =3
4𝑢𝑛 +
1
4𝑢 1 +
1
4Δ𝑡𝑅(𝑢 1 )
𝑢𝑛+1 =1
3𝑢𝑛 +
2
3𝑢 2 +
2
3Δ𝑡𝑅(𝑢 2 )
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CFL condition in 1D for up-to 2nd order spatial discretization: 𝑐Δ𝑡
h≤
1
2𝑝+1,
in which |𝑐| is the wave speed for the given problem, and DG spatial discretization
polynomials of degree 𝑝 = 𝑘 − 1 and RK method of order 𝑘 are used
Higher Order in Space
1. Modal: Solutions are represented by sums of modal coefficients
multiplied by a set of polynomials, e.g., 𝑢 𝑥, 𝑡 = 𝑖=1𝑁 𝑢𝑖 𝑡 𝑃𝑖(𝑥)
– 𝑃𝑖 is often orthogonal polynomials, e.g., Legendre (see left figure below)
2. Nodal: Cells are comprised of multiple nodes on which the solution is
defined. Reconstruction of the cell is then based on fitting an
interpolating polynomial, e.g., 𝑢 𝑥, 𝑡 = 𝑖=1𝑁 𝑢𝑖 𝑡 𝑙𝑖(𝑥)
– 𝑙𝑖 is a Lagrange polynomial (see right figure below)
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Higher Order in Space
For the general case, in which we have 𝑈 ∈ ℝ and 𝑃 ∈ 𝑃𝑘, we get a
(𝑘 + 1) × (𝑘 + 1) system:
(1,1) ⋯ (𝑥𝑘 , 1)⋮ ⋱ ⋮
(1, 𝑥𝑘) ⋯ (𝑥𝑘 , 𝑥𝑘)
𝑎0(𝑡)⋮
𝑎𝑘(𝑡)=
𝑟0⋮𝑟𝑘
𝑝, 𝑞 = 𝐼𝑗
𝑝 𝑥 𝑞 𝑥 𝑑𝑥
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Remember: This system is local
per cell
An orthogonal basis yields a
diagonal matrix
Example of an orthogonal basis in 2D on
triangles (generated with modepy)
Limiters
• For computing solutions with strong discontinuities, it is generally
necessary to apply a limiter to prevent spurious numerical oscillations
• This is typically done in a postprocessing step for DG schemes
• Main idea: Replace the original DG-polynomial in «troubled cells» with
another polynomial of the same order that is less oscillatory than the
original
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After applying limiter
Original DG-polynomial
Limiters
• The original cell average should be maintained for conservation
• Choice of limiter is highly problem dependent
• Builds upon knowledge from developing slope limiters for FVM
• For example: Minmod, generalized minmod, WENO
• See, e.g., A simple weighted essentially nonoscillatory limiter for Runge-Kutta
discontinuous Galerkin methods, X. Zhong and C.-W. Shu
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Extension to Higher Dimensions
We again start with a scalar conservation law, now in 2D
𝑢𝑡 + 𝑓 𝑢 𝑥 + 𝑔 𝑢 𝑦 = 0
Multiply with a test function 𝑣 and integrate over some cell
𝐼𝑗𝑘 = [𝑥𝑗−
1
2
, 𝑥𝑗+
1
2
] × [𝑦𝑘−
1
2
, 𝑦𝑘+
1
2
]
𝐼𝑗𝑘
𝑢𝑡𝑣 + 𝑓 𝑢 𝑥𝑣 + 𝑔 𝑢 𝑦𝑣𝑑𝑥 = 0
Then integrate by parts
𝐼𝑗𝑘
𝑢𝑡𝑣𝑑𝑉 − 𝐼𝑗𝑘
[𝑓 𝑢 , 𝑔(𝑢)] ∗ 𝛻𝑣 𝑑𝑉 + 𝜕𝐼𝑗𝑘
𝒏 ∗ [𝑓 𝑢 , 𝑔(𝑢)] 𝑣 𝑑𝑆 = 0
…and follow the derivation of the 1D scheme
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Extension to Higher Dimensions
𝐼𝑗𝑘
𝑢𝑡𝑣𝑑𝑥 − 𝐼𝑗𝑘
[𝑓 𝑢 , 𝑔(𝑢)] ∗ 𝛻𝑣 𝑑𝑉 + 𝜕𝐼𝑗𝑘
𝒏 ∗ [𝑓 𝑢 , 𝑔(𝑢)] 𝑣 𝑑𝑆 = 0
Use a suitable quadrature rule (e.g., Gauss)
𝐼𝑗𝑘
[𝑓 𝑢 , 𝑔(𝑢)] ∗ 𝛻𝑣 𝑑𝑉 =
𝑚=0
1
𝑛=0
1
𝜔𝑚𝑛 [𝑓 𝑢𝑚𝑛 (𝑣𝑥)𝑚𝑛 + 𝑔 𝑢𝑚𝑛 (𝑣𝑦)𝑚𝑛 ]
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𝑁(𝑢) 𝑆(𝑢)
Parallel Computation
• DG almost perfectly parallel
• Divide into three computational kernels:
𝑑𝑢
𝑑𝑡= 𝑅 𝑢 = 𝑁 𝑢 + 𝑆(𝑢− , 𝑢+ )
• GPU Accelerated Discontinuous Galerkin Methods for Shallow Water
Equations, R. Gandham, D. Medinay, and T. Warburton
• Parallel Implementation of the Discontinuous Galerkin Method, A. Baggag, H.
Atkins, and D. Keyes
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3. Surface integral
2. Volume integral
1. Time integration
(solve ODEs)
𝑢− 𝑢+
Parallel Computation
• Surface integrals depends only on neighboring cells, regardless of the
order of the scheme
• For a structured 2D grid this leads to the following data dependency,
assuming a block decomposition of the domain:
• Very similar to 1st order finite volume methods
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Stencil
Apron
Parallel Computation
• FVM (FDM)
– Special treatment of
boundaries necessary
– Store one value per cell
(per conserved variable)
– High-order solution
requires reconstruction
larger stencil
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• DG
– Easy boundary
conditions (choose a
suitable boundary flux)
– The higher DOF, the
more values must be
stored per cell
– High-order solution in
each cell
Parallel Computation
• New multi-core and many-core architectures can greatly speed up
performance for parallel algorithms
• Since explicit DG is an embarrassingly parallel algorithm, it should run
very efficiently on these architectures
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Architecture
Domain
decomposition
Multi-core CPUs Xeon Phi GPUs
• National Center for Atmospheric Research’s (CISL’s Technology Development Division)
evaluation of accelerator technology within existing weather and climate model codes
• DG_KERNEL benchmark performs a gradient operation from the Discontinuous
Galerkin version of the High Order Methods Modeling Environment (HOMME)
• Specialized versions of the kernel were written for each target architecture (in
cooperation with Intel for the Sandy Bridge CPU and the Xeon Phi)
From 2013 CISL Annual Report
PDE Frameworks Supporting RKDG
• Diffpack– Non-free license
– Collection of C++ libraries and utility programs
– Mature software & large community of users
– Visualization: vtk, matlab, gnuplot, and more
• Dune– GPL 2-licensed (free)
– C++ libraries with separation of data structures
and algorithms by abstract interfaces
– Mature software & large community of users
– Visualization: vtk, GRAPE
• Hedge– MIT licensed (almost free)
– Focus: Fast and easy to use
– Python «front-end» and C++ «back-end»
– Can run on CUDA-capable graphics card
– Visualization: vtk, silo
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Goals for Numerical Methods
1. Sufficient accuracy
– High order and/or high grid resolution
– Does the solution match analytical solutions? (Verification)
– Does the solution replicate experimental data? (Validation)
2. Flexibility
– Make solvers as generic as possible
– Solve classes of problems using the same code
3. Robustness
– Can we always expect a solution from our model?
– Will it break down for, e.g., discontinuities or wet/dry-interfaces?
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Goals for Numerical Methods
4. Efficiency & Scaling
– How long does it take to compute a solution?
– Convergence rate?
– Consider computational platform: Desktop PC vs. MPI and supercomputer
5. Easy problem prototyping and code maintenance
– Avoid ad-hoc solutions
– Should be easy to set up a simulation (e.g., few lines of code)
• Choice of method is often dictated by physical problem, domain
complexity, required levels of accuracy, and available
computational resources
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Summary
• Hyperbolic problems
– Arise in a multitude of different scientific areas
– Typically form discontinuities – need to handle without smoothing
the solution
– Finite speed of propagation, which enables explicit time stepping
• Discontinuous Galerkin
– Locality of FVM combined with variable DOF per element
– General and flexible framework for solving large classes of PDEs
– Conceptually no difference between 1D, 2D or N-D
– Support for locally adaptive numerical solutions (ℎ𝑝-adaptivity),
and meshes can be both non-conforming and unstructured
– The method is highly suitable for parallel computations – only
depends on nearest neighbors in grid
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Further Reading
• Book by Hesthaven and Warburton
• PhD-course slides by Hesthaven
(inspired some of my slides)
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Further Reading
• Essay for a wider audience: – COCKBURN, B: Discontinuous Galerkin Methods
• Runge-Kutta Discontinuous Galerkin Method:– COCKBURN, B.; SHU, C.-W., et al.: The Runge-Kutta discontinuous Galerkin method for
conservation laws. Series of articles I–V.
– COCKBURN, B.: Discontinuous Galerkin methods for convection-dominated problems. In: BARTH,
T.; DECONINK, H. (eds.): High-Order Methods for Computational Physics. Lecture Notes in
Computational Science and Engineering, 9. Springer Verlag, 1999, pp. 69–224.
• Development and state-of-the-art pre-1999: – COCKBURN, B.; KARNIADAKIS, G.; SHU, C.-W.: The development of discontinuous Galerkin
methods. In: COCKBURN, B.; KARNIADAKIS, G.; SHU, C.-W. (eds.): Discontinuous Galerkin
Methods. Theory, Computation and Applications. Lecture Notes in Computational Science and
Engineering, 11. Springer Verlag, February 2000, pp. 3–50.
• Updated review:– COCKBURN, B.; SHU, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-
dominated problems. J. Sci. Comput., 16 (2001), 173–261.
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